Optimal. Leaf size=16 \[ \frac{1}{28} x^{28} \left (b+c x^2\right )^{14} \]
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Rubi [A] time = 0.0541807, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {1584, 446, 74} \[ \frac{1}{28} x^{28} \left (b+c x^2\right )^{14} \]
Antiderivative was successfully verified.
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Rule 1584
Rule 446
Rule 74
Rubi steps
\begin{align*} \int x \left (b+2 c x^2\right ) \left (b x^2+c x^4\right )^{13} \, dx &=\int x^{27} \left (b+c x^2\right )^{13} \left (b+2 c x^2\right ) \, dx\\ &=\frac{1}{2} \operatorname{Subst}\left (\int x^{13} (b+c x)^{13} (b+2 c x) \, dx,x,x^2\right )\\ &=\frac{1}{28} x^{28} \left (b+c x^2\right )^{14}\\ \end{align*}
Mathematica [B] time = 0.0058507, size = 182, normalized size = 11.38 \[ \frac{13}{4} b^2 c^{12} x^{52}+13 b^3 c^{11} x^{50}+\frac{143}{4} b^4 c^{10} x^{48}+\frac{143}{2} b^5 c^9 x^{46}+\frac{429}{4} b^6 c^8 x^{44}+\frac{858}{7} b^7 c^7 x^{42}+\frac{429}{4} b^8 c^6 x^{40}+\frac{143}{2} b^9 c^5 x^{38}+\frac{143}{4} b^{10} c^4 x^{36}+13 b^{11} c^3 x^{34}+\frac{13}{4} b^{12} c^2 x^{32}+\frac{1}{2} b^{13} c x^{30}+\frac{b^{14} x^{28}}{28}+\frac{1}{2} b c^{13} x^{54}+\frac{c^{14} x^{56}}{28} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.004, size = 157, normalized size = 9.8 \begin{align*}{\frac{{c}^{14}{x}^{56}}{28}}+{\frac{b{c}^{13}{x}^{54}}{2}}+{\frac{13\,{b}^{2}{c}^{12}{x}^{52}}{4}}+13\,{b}^{3}{c}^{11}{x}^{50}+{\frac{143\,{b}^{4}{c}^{10}{x}^{48}}{4}}+{\frac{143\,{b}^{5}{c}^{9}{x}^{46}}{2}}+{\frac{429\,{b}^{6}{c}^{8}{x}^{44}}{4}}+{\frac{858\,{b}^{7}{c}^{7}{x}^{42}}{7}}+{\frac{429\,{b}^{8}{c}^{6}{x}^{40}}{4}}+{\frac{143\,{b}^{9}{c}^{5}{x}^{38}}{2}}+{\frac{143\,{b}^{10}{c}^{4}{x}^{36}}{4}}+13\,{b}^{11}{c}^{3}{x}^{34}+{\frac{13\,{b}^{12}{c}^{2}{x}^{32}}{4}}+{\frac{{b}^{13}c{x}^{30}}{2}}+{\frac{{b}^{14}{x}^{28}}{28}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.13387, size = 211, normalized size = 13.19 \begin{align*} \frac{1}{28} \, c^{14} x^{56} + \frac{1}{2} \, b c^{13} x^{54} + \frac{13}{4} \, b^{2} c^{12} x^{52} + 13 \, b^{3} c^{11} x^{50} + \frac{143}{4} \, b^{4} c^{10} x^{48} + \frac{143}{2} \, b^{5} c^{9} x^{46} + \frac{429}{4} \, b^{6} c^{8} x^{44} + \frac{858}{7} \, b^{7} c^{7} x^{42} + \frac{429}{4} \, b^{8} c^{6} x^{40} + \frac{143}{2} \, b^{9} c^{5} x^{38} + \frac{143}{4} \, b^{10} c^{4} x^{36} + 13 \, b^{11} c^{3} x^{34} + \frac{13}{4} \, b^{12} c^{2} x^{32} + \frac{1}{2} \, b^{13} c x^{30} + \frac{1}{28} \, b^{14} x^{28} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 0.897156, size = 402, normalized size = 25.12 \begin{align*} \frac{1}{28} x^{56} c^{14} + \frac{1}{2} x^{54} c^{13} b + \frac{13}{4} x^{52} c^{12} b^{2} + 13 x^{50} c^{11} b^{3} + \frac{143}{4} x^{48} c^{10} b^{4} + \frac{143}{2} x^{46} c^{9} b^{5} + \frac{429}{4} x^{44} c^{8} b^{6} + \frac{858}{7} x^{42} c^{7} b^{7} + \frac{429}{4} x^{40} c^{6} b^{8} + \frac{143}{2} x^{38} c^{5} b^{9} + \frac{143}{4} x^{36} c^{4} b^{10} + 13 x^{34} c^{3} b^{11} + \frac{13}{4} x^{32} c^{2} b^{12} + \frac{1}{2} x^{30} c b^{13} + \frac{1}{28} x^{28} b^{14} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.156746, size = 182, normalized size = 11.38 \begin{align*} \frac{b^{14} x^{28}}{28} + \frac{b^{13} c x^{30}}{2} + \frac{13 b^{12} c^{2} x^{32}}{4} + 13 b^{11} c^{3} x^{34} + \frac{143 b^{10} c^{4} x^{36}}{4} + \frac{143 b^{9} c^{5} x^{38}}{2} + \frac{429 b^{8} c^{6} x^{40}}{4} + \frac{858 b^{7} c^{7} x^{42}}{7} + \frac{429 b^{6} c^{8} x^{44}}{4} + \frac{143 b^{5} c^{9} x^{46}}{2} + \frac{143 b^{4} c^{10} x^{48}}{4} + 13 b^{3} c^{11} x^{50} + \frac{13 b^{2} c^{12} x^{52}}{4} + \frac{b c^{13} x^{54}}{2} + \frac{c^{14} x^{56}}{28} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.08891, size = 211, normalized size = 13.19 \begin{align*} \frac{1}{28} \, c^{14} x^{56} + \frac{1}{2} \, b c^{13} x^{54} + \frac{13}{4} \, b^{2} c^{12} x^{52} + 13 \, b^{3} c^{11} x^{50} + \frac{143}{4} \, b^{4} c^{10} x^{48} + \frac{143}{2} \, b^{5} c^{9} x^{46} + \frac{429}{4} \, b^{6} c^{8} x^{44} + \frac{858}{7} \, b^{7} c^{7} x^{42} + \frac{429}{4} \, b^{8} c^{6} x^{40} + \frac{143}{2} \, b^{9} c^{5} x^{38} + \frac{143}{4} \, b^{10} c^{4} x^{36} + 13 \, b^{11} c^{3} x^{34} + \frac{13}{4} \, b^{12} c^{2} x^{32} + \frac{1}{2} \, b^{13} c x^{30} + \frac{1}{28} \, b^{14} x^{28} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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